Completeness of Exponentials and Beurling's Theorem on $\mathbb{R}^n$ and $\mathbb{T}^n$

TL;DR

This paper extends Beurling's theorem to multiple dimensions, linking the non-vanishing of Fourier transforms of measures with exponential density problems on $ ^n$ and $ ^n$, and introduces new related results.

Contribution

It provides a several variable analogue of Beurling's theorem, establishes equivalences between different problems, and generalizes the theorem to higher-dimensional tori.

Findings

Proved a multi-dimensional version of Beurling's theorem.

Established an equivalence between Fourier transform non-vanishing and exponential density problems.

Generalized Beurling's theorem to $ ^n$.

Abstract

A classical result of Arne Beurling states that the Fourier transform of a nonzero complex Borel measure $\mu$ on the real line cannot vanish on a set of positive Lebesgue measure if $\mu$ has certain decay. We prove a several variable analogue of Beurling's theorem by exploring its connection with the well-known problem concerning the density of linear span of exponentials in a certain weighted normed linear space of continuous functions. In the process, we also prove some new results of this genre and establish an equivalence between the above two problems. We also obtain a generalisation of Beurling's theorem and prove these results on the $n$-dimensional torus $\mathbb{T}^n.$